Mathematics developed early in India. After the time of the ancient scriptures and ritual texts, a new period of mathematical and astronomical activity took place, starting in the fifth century A.D. One of the first astronomical treatises from this period is the Āryabhatīya, by Āryabhata, on which Bhāskara I wrote a commentary. The mathematical chapter of this commentary is the subject of Expounding the Mathematical Seed .

The book consists of two volumes. One volume contains an introduction to the mathematics of the text and Bhāskara's way of interpreting this mathematics, followed by a literal translation of the commentary by Bhāskara. The other volume contains Keller's supplements, which include mathematical and linguistic explanations of the text, an appendix on Indian astronomy and a very useful glossary.

The translation starts by defining the decimal place-value number system and covers subjects such as cubes and cube roots, similar triangles, 'half-chords' (sines), series and equations in two unknowns. It contains the rules by Āryabatha and explanations and examples by Bhāskara. Keller provides extensive explanations in the supplements, by which Bhāskara's often hardly understandable methods of calculation are transformed into recognizable mathematics, enabling the reader to perform the calculations and enjoy the way mathematics was done by Bhāskara.

What I found hard to grasp is the last and most extensive section of the book, which concerns the so called pulverizer (a method for finding integer solutions of linear equations in two unknowns). Whereas in the other sections the examples given by Bhāskara are of help understanding the theory, the examples in the last section are merely astronomical and difficult to truly understand, even with the explanation in the supplements. However, the pure mathematical method itself is explained thoroughly by Keller, unraveling the different steps in the calculation and relating the different ways of solving the problem.

Reading the book was a great joy to me. The translation is clear, but also literal, which enabled me, with one year of Sanskrit study, to read parts of the Sanskrit text along with it fairly easily. It also helped that the key words have been noted down in the translation and that the glossary is sufficient in almost all of my cases of doubt. In the book itself, however, only the verses by Āryabata have been noted down in Sanskrit (in Roman script), not the commentary by Bhāskara. Nevertheless, since the translation is very literal, Bhāskara’s way of writing can be felt through the translation. And let me be clear: knowledge of any Sanskrit is by no means necessary to read and enjoy this book, which is mainly a text on mathematics.

Altogether, I find the book very complete and readable. Though it focuses on one single text, it gives a clear impression of the way mathematics was done in India at the time of Bhāskara. Along the way, Keller mentions questions about aspects of the original text and its context that remain unanswered. This makes the book especially interesting for anyone interested in the history of mathematics, to read for fun or for study or research purposes.

Aldine Aaten recently completed her M.Sc. in mathematics at Leiden University in the Netherlands. She is now learning Sanskrit and preparing for Ph.D. research in the history of Indian mathematics.